Prime Factors of the Denominator of a Rational Number with Decimal Expansion 327.7081

Video Explanation

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Solution

Question: A rational number in its decimal expansion is 327.7081. What can you say about the prime factors of q when this number is expressed in the form p/q? Give reasons.

Step 1: Observe the Nature of the Decimal Expansion

The given decimal number 327.7081 is a terminating decimal, because it ends after a finite number of digits.

Step 2: Use the Property of Terminating Decimals

A rational number expressed in the form p/q, where p and q are integers and q ≠ 0, has a terminating decimal expansion if and only if the prime factorisation of q contains only the primes 2 and/or 5.

Step 3: Draw the Conclusion

Since 327.7081 is a terminating decimal, the denominator q (in its lowest form) must have no prime factors other than 2 and 5.

Final Answer

∴ The prime factors of q are only 2 and/or 5.

Conclusion

Thus, because the given rational number has a terminating decimal expansion, the denominator q in p/q must be of the form 2^m × 5ⁿ, where m and n are non-negative integers.